Find all values of $x$ that satisfy
\[5x - 1 < (x + 1)^2 < 7x - 3.\]
The left inequality becomes $5x - 1 < x^2 + 2x + 1,$ or
\[x^2 - 3x + 2 > 0.\]This factors as $(x - 1)(x - 2) > 0,$ and the solution to $x \in (-\infty,1) \cup (2,\infty).$

The right inequality becomes $x^2 + 2x + 1 < 7x - 3,$ or
\[x^2 - 5x + 4 < 0.\]This factors as $(x - 1)(x - 4) < 0,$ and the solution is $x \in (1,4).$

The intersection of $(-\infty,1) \cup (2,\infty)$ and $(1,4)$ is $\boxed{(2,4)}.$